IOP PUBLISHING SMART MATERIALS AND STRUCTURES

Smart Mater. Struct. 18 (2009) 104001 (13pp) doi:10.1088/0964-1726/18/10/104001

Porosity tailoring mechanisms in sonicatedpolymeric foamsC Torres-Sanchez and J R Corney

Department of Design, Manufacture and Engineering Management, University of Strathclyde,James Weir Building, Glasgow G1 1XJ, UK

E-mail: carmen.torres@strath.ac.uk

Received 15 March 2009, in final form 8 June 2009Published 10 September 2009Online at stacks.iop.org/SMS/18/104001

AbstractFunctionally graded cellular microstructures whose porosity (i.e. volume fraction of void tosolid) is engineered to meet specific requirements are increasingly demanded by bio-engineers,who wish to exploit their synergistic mechanical, chemical and thermal properties. Becausetraditionally polymeric foams have been manufactured with homogeneous porosity, establishedprocesses cannot control the distribution of porosity throughout the resulting matrix. Motivatedby the creation of a flexible process for engineering heterogeneous foams, this paper reportshow the manufacture of polymeric foams with a variable porosity distribution can be achievedby ultrasound irradiation during the ‘sensitive’ stages of the polymerization reaction. This paperreports how for each of the five distinctive stages of polymerization (i.e. cream, rising, packing,gelation and solidification) the energy and mass balances were studied in order to determine theunderlying mechanisms that ultrasound employs to affect the reaction. It was concluded thatcontrolled ultrasonic irradiation affects convective mass transfer during foaming, especiallyduring ‘rising’ and ‘packing’ stages, and enhances the diffusion of the blowing agent(i.e. CO2(g)) from bubble to bubble in the ‘packing’ and ‘gelation’ stages. The mechanicalwork put into the system by ultrasound assists both the convection and diffusion by increasingthe rate of mass flux. The paper concludes with some experimental results that support theabove hypotheses.

(Some figures in this article are in colour only in the electronic version)

Nomenclature

A surface area, m2

B mass of blowing agent in the gas phase perunit polymer mass

CB blowing agent molar concentration,kmol m−3

c speed of sound, m s−1

cp specific heat at constant pressure,J kg−1 K−1

DB mass diffusion coefficient for the blowingagent, m2 s−1

dS0 bubble shell thickness at rest, mdB/dt rate of formation of blowing agent, s−1

dX/dt reaction rate, rate of increase of reactionextent, s−1

F frequency, s−1

GS elastic shell parameter (bubble); shearmodulus, Pa

�Hr total heat reaction per unit of polymermass, J kg−1

hv latent formation (vaporization) heat ofblowing agent (solvent) per unit mass,J kg−1

hm convection mass transfer coefficient, m s−1

K thermal conductivity, W m−1 K−1

k wavenumber•

NB molar rate of increase of specie B per unitvolume due to convection, kmol s−1 m−3

P pressure, Pap(t) instantaneous acoustic pressure, PaR bubble radius, mT temperature, Kt time, s

0964-1726/09/104001+13$30.00 © 2009 IOP Publishing Ltd Printed in the UK1

Smart Mater. Struct. 18 (2009) 104001 C Torres-Sanchez and J R Corney

u velocity, m s−1

xB mole fraction of specie B, CB/Cx, y, z Cartesian coordinates, mZ acoustic impedance, kg s−1 m−2 or RaylGreek symbolsα convection coefficient, W m−2

γ polytrophic exponent of the gasμ viscosity, N s m−2

μS viscous shell parameter (bubble),N s−1 m−2

ρ density, kg m−3

ω angular frequency, s−1

ωBo initial fraction of blowing agentSubscripts( )x,y,z conditions at x, y, z( )w conditions at the vessel wall( )0 initial conditions; at rest( )B,S referred to blowing agent on the membrane

surface( )B,∞ referred to blowing agent under free stream

conditions( )A referred to incident acoustic wave( )L referred to liquid( )S referred to the surfaceOverbar averaged values in time; time mean* referred to the foam∇ Laplacian operator

1. Introduction

Nature provides many good examples of heterogeneousmaterials: e.g. bone (figure 1(a)), a calcium phosphate mineralof a functionally graded porosity reinforced by collagen fibrils;wood and plants stems (figure 1(b)), a composite of fibrouscellulose in a matrix of lignin; silk, chains of entangledmonomers. Since the earliest times, humans too havedevised manufacturing processes that allow the advantagesof mixing materials to be exploited. Bricks (made of mudand reinforced with straw for construction purposes), paper(a matrix of cellulose microfibres) or pottery (ceramic utensilsfor cooking whose porous structure permits thermal insulation)are illustrations of the pre-industrial uses of heterogeneousmaterials. More recently, engineers have also recognized thatthe performance of materials can be dramatically improved iftheir composition and structure are varied to match preciselytheir functional requirements [1]. Such heterogeneousmaterials have engineered gradients of composition, orstructure, which offer superior performance over traditionalhomogeneous materials. Indeed, heterogeneous materialsfrequently demonstrate dramatic synergy, with their overallperformance being far greater than a straightforward sum ofthe individual constituents. These types of materials offergreat promise in fields where a high performance technologyor active functionality is required (e.g. biomaterials, aerospace)because their nature offer the possibility of a compositionwhere different substances can be blended, mixed, shaped orassembled to form components for optimal performance.

Figure 1. Cross-sections (a) bone (courtesy Dr A MacRae, Univ ofCalgary, Canada), (b) bamboo stem—obtained by CPD, (courtesyQuorumTech Ltd), (c) polymeric foam irradiated at 20 kHz and26 000 Pa, 3.70 cm from probe, (d) polymeric foam irradiated at30 kHz and 8900 Pa, 4.90 cm.

A polymeric foam is a particular example of aheterogeneous material, since it is composed of at least twophases, one (or more) solid, plus voids whose size anddistribution can be varied. Polymeric foam materials havedemonstrated great application potential in a myriad of fields(biomaterials, tissue engineering, structural mechanics, etc)because of their lightness, low density, chemical inertness, highwear resistance, thermal and acoustic insulation [2]. This kindof versatility makes foam exceptional as a design material.Moreover, they have compositional similarities with naturalbone and, some types, a certain level of bioresorbability. Foamcore materials offer weight minimization, and the possibilityof being blended with ceramic, or metal, to form polymer-ceramic/metal composites that overcome the disadvantages ofa pure polymeric foam artefact (e.g. poor mechanical strength,short-lived nature, rapid degradability, etc).

The structure of a foam is characterized by thedistribution, size and wall thickness of cells in the bulkmaterial. That distribution has a direct correlation with themechanical properties of the solid foam. Therefore, whena foamed material’s behaviour (e.g. sustained vibrations orimpacts) needs to be engineered, its cellular structure isan obvious starting point. Consequently a reliable porositydistribution representation and measurement method is veryimportant as it will allow a feedback loop to the manufacturingconditions in order to obtain a cellular architecture with theenhanced mechanical, chemical and/or thermal performanceintended.

The aim of this paper is to report work which hasdemonstrated that positioning a foaming polymeric matrix

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Smart Mater. Struct. 18 (2009) 104001 C Torres-Sanchez and J R Corney

within an ultrasonic field (i.e. with known frequencyand acoustic pressure amplitude) during the polymerization‘sensitive stages’, permits the tailoring of the distribution ofbubbles (i.e. cells) to a desired size [3] (figures 1(c) and (d)).‘Sensitive stages’ are those phases during polymerization inwhich ultrasound can have an observable impact becauseevents such as diffusion or convection are predominant.Therefore local alteration in the gas/vapour production andexchange among bubbles can provoke changes in the foam’sfinal porosity.

This paper is structured as follows: after briefly reviewingthe literature concerning foam chemistry (i.e. polymerizationstages and their correspondent energy and mass balances),ultrasound and bubble dynamics in a viscous fluid (section 2),the paper introduces the experimental procedure for a seriesof experiments performed to investigate the effect of anultrasonic field on a vessel filled with polyurethane foamingreactants (section 3). Section 3.1 reports the novel methodused to quantify porosity gradation in the irradiated foams.The following section (section 4) presents the comparisonmade between experimental and simulated results for theacoustic pressure’s effects on the porosity gradation withinthe foam cellular structure. An appraisal of this techniqueas a manufacturing technology for foams with a tailoredporosity distribution is discussed in the final section beforesome conclusions are drawn on the wider significance of thefindings.

2. Background

This section provides the necessary background on thechemistry and physics of the sonication process described later.

2.1. Polymeric foams

Foam is the dispersion of a gas in a liquid, which creates acharacteristic structure when the matrix solidifies. Once cured,the foam consists of individual cells, or pores, the walls ofwhich have completely polymerized and solidified to form askeletal structure. For some polymeric foams, there mightexist a latter stage at which those walls break, leaving anopen structure of interconnected pores (flexible complexion).However, the polyurethane formulation used in this study wassuch that it produced a final close-celled structure after curing(rigid structure) [4]. The chemical reaction that occurs betweenpolyols and diisocyanate group to produce polyurethane [5, 6]with distilled water employed as a blowing agent is:

HO–R–OH(polyol) + O=C=N–R′–N=C=O

(diisocyanate group) → –O–R–O–CO–NH–R′–NH–

CO–(PU) + CO2(gas).

The water diffuses between the chains of polyurethane(PU) reacting at the same time with the isocyanate groups atthe end of the chains, causing the reticulation, or cross-linking,of the polymer, and forming a rigid solid.

2.2. Overall foaming process

A qualitative description of the foaming process usinga chemical blowing agent can be expressed in terms ofcharacteristic stages [5, 7–9, 4] and the events that takeplace in each of them. In order to assess the impact thatultrasonic irradiation might have in these stages, both energyand mass balances are performed. The general equation forthe energy balance is obtained by applying the first law ofthermodynamics to the polymerization, adapted from [10]and [11]: the rate of energy accumulation in a control volumeadded up to the net transfer of energy by fluid flow must beequal to the sum of: the rate of internal heat generation dueto the chemical reaction subtracting the net heat of transferby convection, the net heat transferred by conduction (ruledby Fourier’s law) and the net rate of heat transfer fromthe control volume to its environment due to formation ofthe blowing agent (CO2 from water) and/or vaporization ofany solvent present in the mixture. The overbar refers toaveraged values in time. Limitations exist in this energybalance (i.e. Fourier’s law is not an expression that may bederived from first principles, but a generalization based onexperimental evidence). However, the presumption that theheat flux is normal to an isotherm and happens in the directionof decreasing temperature (i.e. Fourier’s law) is a necessarysimplifying assumption in order to express the heat generationand transfer within this complex system.

ρcp

(∂T

∂ t︸︷︷︸Rate energy accumulation

+ ux∂T

∂x+ uy

∂T

∂y+ uz

∂T

∂z︸ ︷︷ ︸Net energy transfer

)

= (1 − ωBo)�HrρdX

dt︸ ︷︷ ︸Heat produced due to the exothermic reaction

− αA∇(Tx − Tw)︸ ︷︷ ︸Convection

− K∇2T︸ ︷︷ ︸Conduction

− ρhvdB

dt︸ ︷︷ ︸Formation of blowing agent

. (1)

The mass transfer for this particular case combines grossfluid motion (convection) with diffusion (ruled by Fick’s law)to promote the transport of the blowing agent formed bythe reaction, for which there exists a concentration gradient,driving force for the flux of mass within the polymeric matrix.The transfer rate through the polymeric wall (from bubbleto bubble or from cavity to cavity) has been investigatedby studying the concentration boundary layer [10, 11]. Theboundary layer plays an important role in the foaming ofthe polymeric melt process because this manifests via threephenomena: surface friction (i.e. gas/vapour on bubble shelllayer), convection heat transfer and convection mass transferacross the bubble shell layer (both from liquid to bubble, orvice versa). For compressible fluids (e.g. CO2 and vapour)being transferred from a liquid and liquid-to-viscous mediato/from the bubble, the mass balance needs to be done ona molar concentration basis, as density varies within the

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Smart Mater. Struct. 18 (2009) 104001 C Torres-Sanchez and J R Corney

polymeric bulk:

∇2 (C DBxB)︸ ︷︷ ︸Diffusion

+ •N B︸︷︷︸

Convection

=

Concentration rate ofblowing agent in mixture︷︸︸︷

∂CB

∂ t. (2)

This general expression indicates that there are twocontributions to the absolute flux of species (i.e. how theconcentration of blowing agent changes in the mixture fromthe start of the reaction): a contribution due to diffusion(i.e. motion of gas/vapour relative to the motion of mixture)and a contribution due to convection (i.e. motion of gas withinthe mixture). No further simplifications can be made, asthe blowing agent and vapour are compressible fluids andneither the blowing agent diffusion coefficient nor molarconcentration are kept constant throughout the process andwithin the polymeric solution or, at a later stage, solid matrix.

Fick’s law is an analogous expression to Fourier’s law,i.e. it is an empirical relationship. The presumption madein this case relates to the mass transfer resulting froma concentration gradient, and additional effects such astemperature or pressure gradients are considered negligible.The authors believe this is a credible assumption because thespecies concentration gradient is likely to be the dominantdriving potential in most stages of this polymerization reaction.

2.3. Individual stages of polymeric foam formation

Once the general equations for both energy and mass transferhave been defined in the context of polymeric foaming,each of the individual stages will be described below andexplored in terms of macroscopic behaviour and mass/heattransfer. The five main stages in the polyurethane foaminghave been established as: cream, rising, packing, gelation andsolidification stages [12].1. Cream stage. Bubble nucleation occurs upon the additionof the catalyst (i.e. water). The carbon dioxide gas, CO2,produced acts as the blowing agent and gives a cloudy, creamyappearance to the mixture. The mass transfer regarding theblowing agent is null, as its concentration gradient is zero atthis early instant. The energy balance main contributions arefrom the exothermic chemical reaction and the formation ofthe blowing agent.

ρcp

(∂T

∂ t+ ux

∂T

∂x+ uy

∂T

∂y+ uz

∂T

∂z

)

= (1 − ωBo)�HrρdX

dt− ρhv

dB

dt. (3)

As the foam growth has not started yet, the initialdimensions of the control volume are small. In addition itcan be assumed that the temperature gradient in x, y, z isnegligible, as the exothermic reaction and formation of theblowing agent are homogeneous throughout the mixture. Thissimplifies the energy balance at this stage.

∂T

∂x= ∂T

∂y= ∂T

∂z∼= 0 (4)

ρcp∂T

∂ t= (1 − ωBo)�Hrρ

dX

dt− ρhv

dB

dt. (5)

2. Rising stage. After nucleation, the polymeric mass beginsits free expansion in an open vessel (i.e. at constant pressurebut with variable volume). Due to the exothermic nature ofthe polymeric reaction, the temperature inside the containeris greater than the temperature at its walls. A gradient oftemperature is established. During this stage the liquid foamis a metastable system that evolves dynamically due to twoprocesses: foam drainage (liquid flows through the interstitialvolume between bubbles) and foam coarsening (gas exchangebetween bubbles). As the mixture is still a liquid, convectionwill be the main heat transfer agent within. At this specificstage, the contributions to the energy balance are: the heatgenerated due to the reaction, convection and formation ofblowing agent (CO2 gas). The assumptions at the ‘rising’ stageare the following:

• Velocities ux and uy can be considered of equal value dueto the small dimensions of the container. In addition, aplane front is moving faster in the z-direction due to therapid growth of the foam.

• Likewise, the gradient of temperature in the x- and y-direction is small compared to the one in the z-directionbecause the rising of the foam is only in the z-direction.Temperature at the wall (Tw) stays below the temperaturein the vessel (reaction temperature, Tx,y ) so:

∂(Tx − Tw)

∂x= ∂(Ty − Tw)

∂y= 0. (6)

Therefore, the general equation can be simplified asfollows:

ρcp

(∂T

∂ t+ ux

∂T

∂x+ uy

∂T

∂y+ uz

∂T

∂z

)

= (1 − ωBo)�HrρdX

dt− αA

∂(Tz − Tw)

∂z− ρhv

dB

dt. (7)

As the mass transfer is predominantly convective at thisstage, the general equation for the mass balance can be writtenas,

∂CB

∂ t≈ •

N B (8)

∂CB

∂ t≈ d

dt

(hm As

(CB,S − CB,∞

)). (9)

3. Packing stage. The CO2 generated raises the polymeric meltuntil a maximum height is reached. The exothermic energyof the reaction is used to entangle units that will form cells.The formed cells have trapped CO2 gas (the blowing agentthat is formed during the reaction) inside. This gas diffusesthrough the polymer melt into the bubbles and moves from cellto cell, these might coalescence and/or implode, and this canprovoke events of partial collapse of the foam. This stage is theone at which more phenomena contribute in terms of energytransfer: chemical reaction, formation of CO2, convectionand conduction, due to the viscoelastic characteristics of themixture. The following assumption is introduced into thegeneral energy balance:

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Smart Mater. Struct. 18 (2009) 104001 C Torres-Sanchez and J R Corney

• Except at localized areas where events of collapse mighthappen during a short period of time, velocity on the z-coordinate can be considered zero, as the foam does notgrow (uz = 0).

ρcp

(∂T

∂ t+ ux

∂T

∂x+ uy

∂T

∂y

)

= (1 − ωBo)�HrρdX

dt− αA∇(Tx − Tw)

− K∇2T − ρhvdB

dt. (10)

Along with the energy balance, a mass balance needs tobe considered for the reaction from the start of the ‘packing’stage to the instant at which ‘gelation point’ takes place.This mass balance is of a two fold nature: diffusive andconvective, and both elements have to be considered whenrecalling the general equation for mass transfer and expandingits terms, as shown below. It also needs to be noticed that theassumptions taken in order to simplify equation (2) into (11)are that the gradient in the three coordinates (x, y and z) ofspecie B, i.e. blowing agent, ∇xB, is much greater than thegradient of the concentration, ∇C , as the viscoelastic mixtureis of homogeneous nature, i.e. same chemical compositionat any point of the mass, and than any gradient of diffusioncoefficient, ∇D, this considered negligible.

∂

∂x

(C DB

∂xB

∂x

)+ ∂

∂y

(C DB

∂xB

∂y

)

+ ∂

∂z

(C DB

∂xB

∂z

)+ ∂

∂ t

(hm As

(CB,S − CB,∞

))

= ∂CB

∂ t. (11)

From this equation, it can be seen that there is adirect relationship between the variation of the blowingagent concentration with time and the diffusivity coefficient,therefore the mass balance contains a diffusive component.Other authors have also shown a similar relationship [13].4. Gelation stage. At this point, the final structure of thefoam is reached. The rigidity of the matrix is high enough toconsider the bubbles as cells. Bubble size becomes fixed andthere is no longer the possibility of the bubbles expanding orcollapsing, as the increasing viscosity of the plastic makes thewalls stiff and strong against shear forces. The contributionsto the energy balance at this stage are: heat generated bythe reaction, conduction and convection. Although convectionis still considered here (due to the presence of large pores),conduction is a dominant phenomenon over convection, whichwill decrease in value until becoming negligible at the start ofthe next stage. The assumptions at this stage are:

• Vapour/gas molecules velocities are considered small anddecreasing at this stage (ux , uy , uz = 0).

• The gradient of temperature along the z-axis (variationof temperature of the foam with height, dT/dz = 0)is considered negligible, since the heat transfer from thefoam surface to the surrounding air is insignificant (airis a poor conductor of heat and there is no variation ofvelocity or foam height in the z-direction. Dissipated heatby convection is also considered negligible due to the lack

of turbulent air above the free surface of the solidifyingfoam).

Introducing these assumptions into the expression for thegeneral energy balance, the following equation is obtained:

ρcp

(dT

dt

)= (1 − ωBo)�Hrρ

dX

dt

− αA

(∂(Tx − Tw)

∂x+ ∂(Ty − Tw)

∂y

)

− K

(∂2T

∂x2+ ∂2T

∂y2

). (12)

The mass balance also contains a term which representsthe diffusion (by Fick’s law in the gradient at the boundarylayer) and, in a smaller contribution, the convection term. Anygradient in the z-coordinate is considered negligible in this casetoo.

∂CB

∂ t= ∂

∂x

(C DB

∂xB

∂x

)+ ∂

∂y

(C DB

∂xB

∂y

)

+ ∂

∂ t(hm As(CB,S − CB,∞)). (13)

After the gelation point, the viscosity increases drasticallyin detriment to the diffusion process, and consequently themass diffusion coefficient (DB) decreases making the transferof the blowing agent slower through the boundary layer, amongwall cavities. The stiffness of the cavities’ walls promotes heattransfer mainly by conduction. As a consequence, it can beconcluded that the soft elastic gel nature created after gelationdoes not promote growth of bubbles.5. Solidification stage. Finally, when all the polymericmass has gelled, the final structure is obtained. Cross-linkingfinishes and foam starts a curing period where cells becomefully solidified. As the foam is rigid and the polymericcells are fully formed, the convection contribution to theenergy balance can be assumed to be zero since the velocityat which vapour/gas molecules move (from cavity to cavityor inside each cavity) is too small to be significant. Thisvelocity decreases rapidly from the gelation point due to theintroduction of a rigidity term in the polymeric matrix. It isalso assumed that the gaseous fluid adheres to the wall of thecavities, and therefore the heat flow at the wall will be byconduction, not by convection [9]. For this reason, heat willbe transmitted mainly by conduction (which causes the dryingof the foam) until there is no temperature gradient in the foam(dT/dt = 0).

ρcp

(dT

dt

)= K

(∂2T

∂x2+ ∂2T

∂y2+ ∂2T

∂z2

). (14)

It is notable that the mass transfer equation for this stagedoes not include a convective term. The only process related tothe mass flux during this stage is the diffusion of any remainingvapour from the foam volume to the outer environment. Thecontent of the blowing agent here is mainly vaporous resultingfrom the drying of the foam.

∇2(C DBxB) = ∂CB

∂ t(15)

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Smart Mater. Struct. 18 (2009) 104001 C Torres-Sanchez and J R Corney

∂

∂x

(C DB

∂xB

∂x

)+ ∂

∂y

(C DB

∂xB

∂y

)

+ ∂

∂z

(C DB

∂xB

∂z

)= ∂CB

∂ t. (16)

2.4. Sonication of multiple bubbles in a viscous matrix

Literature has widely reported ultrasonic irradiation to foamsunder a myriad of specific applications. Examples are:the interaction ultrasound/foam that enabled defoaming inbottling of fizzy drinks and the dissipation of foam in reactionand fermentation vessels [14, 15], controlled polymerizationrate [16], enhancement of contaminants removal [17], aidedfood dehydration [18] and drug delivery [19]. Many of theseapplications exploit the ultrasonically stimulated transient-cavitation effect (rapid growth and explosive collapse ofmicroscopic bubbles). An established research trend focusingon irradiation of foams under stable-cavitation conditions(i.e. rectified diffusion that enlarges the size of the bubble ina sustainable way) has not been found in the literature.

Ultrasound, as any other wave transmitting through amaterial medium, causes the particles of the irradiated mediumto be set into vibrational motion through which they gainkinetic energy [20]. The defining equation for a pulsatingincident field of sound is:

p(t) = P0 + PA cos(ωt + kx). (17)

For the purpose of this work, a wave, which is character-ized by its frequency and amplitude, was continuously emit-ted (standing wave within rigid boundaries, i.e. walls in be-tween which it was transmitting) from the source. Therefore,the defining equation is expressed as [20–22]:

p(t) = P0 + PA cos(kx) sin(2π f t). (18)

Much has been written about motion of a gas bubblein liquid when it is irradiated. Translational motion due tobuoyancy and drag forces are not significant here since theyhave been defined for bubbles in aqueous solutions and in thiscase bubbles are pulsating a polymeric matrix that constrainstheir positions in the matrix from an early stage due to a‘packing effect’ (i.e. bubbles grow competing for space).

In particular, bubble growth rate has been generallypredicted and represented by the Rayleigh–Plesset equationand its several approximations (e.g. Safar’s, Eller’s, Crum’s,etc) [23, 24]. However, this expression was developed for‘free’ bubbles, those which are suspended in a Newtonianliquid (i.e. water). In contrast, the bubbles considered inthis study are so called ‘shelled’ bubbles, bubbles pulsatingin a viscoelastic matrix that offers resistance to the bubble’sexpansion/contraction induced by ultrasonic irradiation. Abubble in a sound field is a highly nonlinear system (i.e. achange in the sound amplitude not only changes the amplitudeof the oscillations, it also changes their shape). When thebubbles are embedded in a viscoelastic polymeric matrix, thisnonlinearity might be reduced due to a more limited amplitudeoscillation. Nonetheless, regardless of its magnitude, the highnonlinearity of the system makes modelling the bubbles motion

and expansion very difficult and some simplifying assumptionsare needed to allow progress in the mathematical description ofthe system.

In the context of this work, the expression for encapsulatedbubble growth rate in a standing wave (i.e. bubbles ina polymeric or starch matrix) by Church [25] and thensimplified by Hoff [26], where the only variable is the bubbleradius, is particularly important. The model is based onspherical bubble growth as a simplification of the bubblegeometry. The modelling of a 3D network of bubbles requiressubstantial work, beyond Hoof’s and Church’s contributions, tocharacterize bubble-to-bubble interactions and their evolutionover time. To date, to the authors’ knowledge no such modelhas been reported.

ρL

(d2 R

dt2R + 3

2

(dR

dt

)2)= P0

((R0

R

)3γ

− 1

)

− p(t) − 4μL

R

dR

dt− 12μSdS0 R2

0

R4

dR

dt− 12GSdS0 R2

0

R3

×(

1 − R0

R

). (19)

When bubbles of initial small radii suffer alternateexpansion/contraction due to the sinusoidal nature of thesound wave field, under conditions of stable-cavitation, thisprocess is positive. Expansions are bigger than contractionsand the bubble growth is in resonance with the sound waveand sustained in time. Bubble dynamics play an importantrole in pore enlargement, but other processes also enhancedby ultrasound (i.e. diffusion and mixing) will influence thedynamics of the process of foam formation. Particularlyimportant in the context of foams and other high viscosityfluids is the ability of ultrasound to produce an increase in masstransport due to diffusion variation [27].

Essentially, equation (19) associates sound and shell prop-erties: sound affects the viscosity of fluids significantly (usu-ally decreasing their viscosity), so acoustic radiation reducesthe diffusion boundary layer, increases the concentration gradi-ent and may increase the diffusion coefficient. In addition, tur-bulent convection provoked by ultrasound decreases the thick-ness of the mass transfer boundary layer, i.e. the wall of thepore, and increases transport through the membrane. However,if the shear forces provoked by ultrasound are excessive, somecells might rupture affecting the viscoelastic equilibrium in thematrix and, in extreme conditions, leading to a foam collapse(effect of transient-cavitation).

The preceding sections have described the chemistry andphysics involved in each stage. These descriptions will berevisited in the discussion when the nature of the sonicationeffect is argued.

3. Methodology

As the energy and mass balances suggest that ultrasonicirradiation might have an impact on those ‘sensitive’polymerization stages (e.g. ‘rising’, ‘packing’ and ‘gelation’),a series of experiments were performed to establish whetherthere was any noticeable effect on the final porosity distributionof polyurethane foam when irradiated while polymerizing. The

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Smart Mater. Struct. 18 (2009) 104001 C Torres-Sanchez and J R Corney

Figure 2. Schematic of the experimental rig, lateral and plan views, showing variable positions of a single foam container.

reactants were irradiated in a temperature controlled (313 K ±1 K) water bath over several values of frequency and acousticpressure. The schematic shown in figure 2 illustrates theultrasonic source and the polypropylene (material chosen forits similar acoustic impedance to water, i.e. a low energy loss atthe interface) container (5 cm diameter, 7 cm height, 0.16 mmthickness). This held the reactants within the water bath, whosewalls were lined to minimize wave reflection. The use of awater bath ensured the temperature of the environment couldbe controlled independently of the effects of ultrasound andavoided overheating due to the continuous functioning of thepiezoelectric probe. The container was firmly clamped witha lab stand and positioned along the longitudinal axis of thebath. The ultrasonic piezoelectric sources used were a 20 kHzBandelin Sonopuls sonotrode, Germany, UW 3200 and a 25and 30 kHz Coltene Biosonic US100, USA. In order to haveboth transducer and foam container aligned, the sonotrode tipwas immersed 2 cm below the free surface, on the same planeas that of the central plane of the container.

The reactants used in this study (Dow Europe GmbH,Switzerland) were pre-treated and the diisocyanate content inthe mixture was rectified to have a fixed 40%. The amountof distilled water added was directly related to that amount(20%vol H2O per ml mixture). This was done using theprocedure of stirring for a standard time of 70 s and minimizingair intake into the mixture. All mixtures were sonicatedin an open-vessel container to avoid the build up of theinternal pressure due to the water vapour and gases (e.g. CO2)generated by the reaction that could provoke unwantedimplosion of bubbles. The containers were perpendicular tothe sonicating probe and had the opposite 180◦ of their surfaceshielded by absorbent material to minimize reflections fromthe walls and enable investigation of the effects by direct ‘nearfield’ sonication. Thermocouples and conductivity probes wereheld in the middle of the mixture and used to monitor thereaction and establish each stage’s completion [12].

The 20 min irradiation period was an off/on cycle of 2 minon/1 min off starting after adding the distilled water, and thenleft in the bath for 30 min until the foam was rigid. Thiscyclic irradiation was established by initial experimentationas sufficient to induce changes in the foam structure without

causing collapse. The ultrasonic irradiation characteristicswere established by previous mapping of the ultrasonic bathusing a needle-type hydrophone (Bruel and Kjær, Denmark,type 8103) shielded with a barrier made of the same open-vessel material for representative values. The sonicationconditions obtained after analysis of the bath dimensions,bath walls behaviour, coupling agent (i.e. water), temperature,etc were used to model the acoustic field (figure 3). Bothacoustic fields (i.e. inside of the water bath and in thecontainers) were modelled using a COMSOL™ Multiphysicsapplication.

The experimental procedure followed is summarized asfollows: (1) a measured amount of reactant was placed in thecontainer located at a certain distance from the sonotrode; (2)the process was initiated by addition of water (the chemicalblowing agent and catalyst for the reaction); (3) ultrasound ofknown acoustic pressure value was applied; (4) on completionof the reaction, the foam was left to cure for 48 h; (5) oncethe sonicated foams were fully cured, they were de-mouldedand cut in half with a coarse-tooth saw and the cross-sectionsscanned for further analysis.

3.1. Quantifying porosity distribution in polyurethane foams

To assess the effects of the ultrasound exposure on thefoam’s cellular structure, a method of characterizing theporosity distribution within a material is essential. Foropen-cell structures (e.g. flexible foams, rocks), porositycan be measured using liquid displacement techniques(e.g. Arquimedes’, toluene infiltration displacement, mercury-porosimetry), which provide an average density value for thebulk material (e.g. measurement permeability and tortuosityin a sample). However, for this work, closed-pore foamswere manufactured and so these methods were not applicable.The lack of a systematic method for the assessment of aheterogeneous material’s porosity [28], was a difficulty fora direct assessment of the cellular structure in the irradiatedfoams. To overcome this obstacle an image processingapplication, known as ‘Topo-porosity mapping’ tool, wasdeveloped in MATLAB™ to allow analysis and delineation ofthe foam porosity observed in a cross-section. This strategyconsidered the density of a cellular solid as the ratio of the

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Figure 3. Acoustic pressure field (volume, left and slice on y axis, right) mapping of the water bath showing partial maxima along the bath,from the probe.

Figure 4. (a) MATLAB™ interface; (b) isoporosity contour lines; (c) and (d) correspondent areas in image analysis and contour lines.

density of the foam (ρ∗) to the density of the solid (ρs)material (ρ∗/ρs) [2]. The density of a foam is indicative ofits porosity. Each sample was sliced and the porosity assessedusing digital image analysis. Similar structure characterizationmethods have been already used in aqueous and polymericfoams [29]. Within the sliced samples, the 3D networkof the foam structure can be clearly observed (figures 1(c)and (d)). The samples were scanned at 1500dpi resolutionin an EPSON Perfection Scanner 1640SU. The purpose of the‘Topo-porosity mapping’ tool was to correlate the topographicdistribution of isolines of density in each sample with themanufacturing process parameters present during its formation(e.g. sonicating irradiation, frequency and relative position inthe acoustic field).

In essence, the program calculated the amount of cellwall material in different cross-sections of the foam. Pointswith the same range of porosity were connected by curvesin the same way that contour lines in a topographic mapconnects continuous points of the same altitude. These

topographic maps of porosity provided information on theporosity distribution within a foam cross-section, indicatingthe relative positions of areas with equivalent porosity(figure 4).

In order to isolate the surface plane, the RGB valuesfor colour of the foam matrix were filtered from the image.Colour power, colour threshold and intensity were also usedto enhance the surface and its contrast. A good contrast inthe image leads to a good qualitative image segmentation.This is a crucial stage in the process as a slight variationin the values may provoke a different porosity result. Inthis case, once the thresholding values are set, these are keptfixed and used for every foam within the batch. Therefore,any possible variation is minimized as each image uses thesame reference value. Using this filtered image, a grid wasapplied to the image, which counted the pixels and adapted(i.e. reduced or expanded) the size of squares in the grid untilthey matched a given value of intensity. This intensity wasset via the mesh spacing initially chosen so it reflected the

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Figure 5. Cross-section of foams sonicated at 20 kHz, same value of acoustic pressure but different positions in the acoustic field (irradiatingprobe was located on the left in the three cases).

observed distribution of cellular porosity. The image wasthen pixelated and each grid contained a value which was thenumber of pixels contained in that area. Applying the ‘contour’option, a set of isoline curves was obtained and connected allpoints with an equal number of pixels, which was indirectlyrelated to porosity and directly linked to density. The resultswere topographic pictures, where points of equal porosity werejoined by contour lines. The relative position of the contourlines gave information about the rate of change within animage. Based on the ratio ρ∗/ρs, the contours effectivelymapped porosity distribution where a value of 200 was set to beequivalent to the density of solid polyurethane. For areas wherethe colour was red, the density was higher, so porosity was low.For areas where colour was blue, porosity was higher. For areaswith no lines, or spaces between lines, there was no variationof porosity in the samples (given the interval used to generatethe plot). For example, when foam occupied 80% of the totalvolume, the value of the lines was 160, as shown in figures 4(c)and (d). By using the same parameters for colour filters andthreshold, a comparative study among samples could be made.A validation of this technique was performed via comparisonof the ‘Topo-porosity’ results versus direct measurements ofporosity on the sample. The description of the procedure andthe results can be found elsewhere [30].

4. Results

The sonicated foams presented a graded porosity and,depending on their position within the acoustic field, thatgradation had different configurations (figure 5). Areas inthe foams’ cross-section with larger pores correspond tothose zones in the bath with higher acoustic pressure values.Therefore, the effect of sonication on the pore architecturecould be directly linked.

The porosity of the irradiated foams was representedusing the isoporosity lines, as explained in the previoussection, which enabled a systematic investigation on howultrasound affected the cellular architecture at each locationin the acoustic field. At the same time, results from theCOMSOL™ simulation of the acoustic field inside of thevessel at each sonication condition (i.e. acoustic pressure andfrequency) were computed. Both results, porosity values fromthe solid samples and acoustic field levels from the sonication

environment were compared against each other. The values ofporosity (figure 6(a) from the MATLAB™ image analysis) andsound pressure (figure 6(b) from the COMSOL™ simulation)along a line through the mid-point (i.e. same depth thanthe immersed sonotrode’s tip) of the container were plotted(figures 6(c) and (d)). This allowed a direct comparisonbetween the porosity gradation measured on the samples’cross-section (figure 6(e)) and the acoustic field that theywere subjected to. For each frequency of irradiation (i.e. 20,25 and 30 kHz), the porosity distribution across the sectionof the foams (solid line) was plotted against the acousticpressure level in the foam container (as extracted fromthe COMSOL™ simulator), assuming the foam’s acousticattenuation to lie between the extremes of water (dash anddot line) and cortical bone (dashed line) (figure 7). Theresults suggested that the samples that were irradiated at higheracoustic pressures presented a better correlation betweenthe porosity distribution and the acoustic pressure level.Likewise, those foams irradiated at lower acoustic energyshowed a weaker correlation with the simulated pressuredistribution.

Although the bulk porosity remained approximately thesame from early stages of the polymerization reaction untilfully cross-linking of the polymer, the local porosity and,therefore, the acoustic impedance, varied continuously. Theacoustic impedance of a viscous fluid is a function of thedensity of the fluid, its viscosity and the frequency ofthe ultrasonic wave [31]. During foam cross-linking, theirradiated medium was a mixture of water, carbon dioxideand polyurethane foam. Therefore, the acoustic impedancewas expected to change from an initial value similar to water(Zwater = 1.48 MRayl), through an acoustic impedance similarto resin (Z resin = 1.5–1.8 MRayl) [32] when the viscosity washigh, evolving finally towards values associated with porousmaterials (7.4–10 MRayl) [21] or compact bone (9.3 MRaylfor a density of 1930 kg m−3) [33] when the foam was fullycured and dry. For the purpose of the irradiated foam in thesimulated bath, the working acoustic impedances that wereused corresponded to the water (Z = 1.48 MRayl; density1000 kg m−3, longitudinal sound velocity c = 1480 m s−1)and to typical trabecular bone (Zbone = 2.6 MRayl for adensity of 1630 kg m−3, c = 1550 m s−1) [34], whichmatched the expected density of the foam at those stages inthe reaction.

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Figure 6. Procedure for analysis of foam irradiated at 25 kHz and 8.85 cm distant from the sonotrode while immersed in the water bath.(a) Isoporosity contours from ‘Topo-porosity’ applied to the cross-section of a foam sample. (b) COMSOL™ simulation of the acousticpressure distribution in a vertical plane from the modelled vessel immersed in the water bath. (c) Porosity values (inverse of density values)extracted along mid-line (AA′) of ‘sonication plane’ aligned with sonotrode. (d) Simulated sound pressure levels extracted along mid-line(BB′) of the ‘sonication plane’ aligned with sonotrode for two acoustic impedances (Z = 1.48 MRayl is water and Z = 2.6 MRayl is bone).(e) Comparison porosity (experimental) versus sound pressure distributions (simulation) for irradiated foam.

5. Discussion

5.1. ‘Sonication window’ for the polyurethane polymerization

The results show a direct correlation between porosity valuesand the acoustic pressure applied to the vessel during foaming.The foams were subjected to a sinusoidal wave in theacoustic field that attenuated in a complex fashion whentravelling through the vessel where the polyurethane mixturewas reacting. It is apparent even to the naked eye thatthe porosity gradation in the foam final structure also showsa parallel topology to that of the standing wave in theacoustic field within the vessel. The physical phenomenaunderlying these results can be visualized with a schematic(figure 8) that illustrates the different ways acoustic cavitationinfluences the size of bubbles in polymeric foams dependingon the level of acoustic pressure. This sketch completes

the descriptions of other researchers who have observedsituations where gas bubbles submerged in liquid could onlysuffer enlargement [23, 35] (stable cavitation) or implosion(transient cavitation) [14]. Our results showed that, forthe polyurethane foams studied, bubble enlargement wasproportional to the sound pressure when this was larger thana lower threshold value (below which there was no effecton the cellular structure), and lower than an upper thresholdvalue, that provoked homogenization and, at an extreme,collapsing of the polymeric cellular structure through breakingthe polymer chains and implosion of bubbles. Both lower andupper threshold set the boundaries for a ‘sonication window’explored here for this particular polyurethane formulation.It is thought that a similar window exists in other foamingpolymers, and this is a subject of further study in orderto optimize the manufacture and porosity tailoring in thesecellular structures.

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Smart Mater. Struct. 18 (2009) 104001 C Torres-Sanchez and J R Corney

Figure 7. Comparison of porosity and sound pressure distributions for foams irradiated at the following conditions: (a) 20 kHz, 3.70 cm fromprobe irradiating at 18 kPa; (b) 25 kHz, 6.35 cm from probe irradiating at and 12 kPa; (c) 30 kHz, 2.45 cm from probe irradiating at and8.9 kPa; (d) 30 kHz, 4.90 cm from probe irradiating at and 8.9 kPa.

Figure 8. Stages of acoustic cavitation exploited for the tailoring ofpolymeric foams.

5.2. ‘Sensitive’ stages to ultrasound in the polyurethanepolymerization

The energy and mass balance associated with each foamformation stage described in section 2.3 describes howphenomena such as the formation of blowing agent,convection, conduction and diffusion all affect the foamformation. The relative importance of the various parametersvaries from stage to stage.

The ‘cream’ stage is when the reaction starts due to theaddition of the catalyst (i.e. water). The contributions tothe energy balance at this stage (equation (5)) are the heatproduced due to the exothermic reaction and the formation ofCO2 from the water. This can be assumed to remain constantwhen fixed quantities of water and monomers are used. It isunlikely that ultrasound plays an important role at this stage,since the kinetics of the blowing agent formation has a muchlarger inertia than the physical effect that ultrasound mightprovoke in the mixture.

The ‘rising’ stage is characterized by convection causedby a rapid increase of volume at constant pressure. The energybalance associated with this stage (equation (7)) is dominatedby a convective term due to the rapid formation of CO2(g),blowing agent. The mass flux in this stage is predominantlyconvective mass transfer (equation (9)). The ultrasonicirradiation is thought to aid a macroscopic convection of theblowing agent produced and its homogeneous distribution intothe polymeric metastable bulk, thus, enhances the formationand swelling of bubbles at this stage.

The ‘packing’ stage is the one phase in which thelargest number of variables contribute in terms of bothenergy (equation (10)) and mass transfer (equation (11)), and,consequently it is not surprising that this is when the ultrasonicirradiation has its biggest effect on the cellular structure. Atthis stage, the melted polymer fills the container uniformly,and the cross-linking reaction takes place in order to turnbubbles into pores, strengthening the neck of the cavities. Thedensity of each element (dx, dy, dz) changes accordingly tothe formation of CO2(g)—blowing agent—and there is massexchange (vapour, not liquid) with the neighbouring elements.The authors believe that a convective transport of mass andenergy is intensified when ultrasound is applied to the sample.

Another observation about the energy balance for thisstage (equation (10)) is that the conduction term can beaffected and enhanced by ultrasonic radiation. Reported inthe literature [7] and successively referenced in [8] is the factthat the thermal conductivity, K is a function of instant foamdensity, ρt . In other words, the dissipation of energy can beaccelerated by the ultrasonic radiation, by both conduction andconvection. This will decrease the difference in temperature(�T ) at inner locations in the vessel and on the perimeter ofthe sample. A forced dissipation of heat assisted by ultrasounddecreases the temperature gradient at the container walls and

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in turn leads to a minimization of skin formation duringprocessing. This is important in the context of manufacturesince skin formation (i.e. a high density layer of the samematerial) is undesirable in foams for structural applications, asthe skin has very different mechanical properties from the corefoam.

In addition to the convective effects, the mass transferresistance (equation (11)) can be decreased with the applicationof ultrasound, as the mass diffusion coefficient (DB) willincrease. If DB can be increased, then the formation ofthe blowing agent can be enhanced [13], as the diffusioncoefficient is a function of the formation of the blowing agent,dB/dt = F(DB). Conversely, the formation of blowingagent suggests that DB is the variable affected by ultrasound.Therefore, if ultrasound can influence the diffusivity, it can alsoaccelerate the formation of CO2, and vice versa, resulting in areaction that is brought to ‘gelation point’ in a shorter time.The mechanical work put into the system by ultrasound assiststhe diffusion by increasing the rate of mass flux. Bubbles aresubjected to stretching, shearing and compression–expansion(Bjerknes) forces.

The effects of ultrasound during other stages are lesspronounced: at ‘gelation’ stage, ultrasound improves thepermeability of the gas through the cavities walls, althoughits impact on the polymer density is less pronounced thanthe effect produced in the ‘packing’ stage. Convection,even if small compared to conduction, is enhanced too, andboth contribute to a larger permeability. It is also obviousthe ultrasound will have no effect on the foam’s porousarchitecture if irradiated during the ‘solidification’ stage,because the porous architecture is fully set, the walls rigid andthe cavities defined.

In summary: controlled ultrasonic irradiation affectsconvective mass transfer during foaming, especially during the‘rising’ and ‘packing’ stages, and enhances the diffusion ofthe blowing agent (i.e. CO2(g)) from bubble to bubble in the‘packing’ and ‘gelation’ stages.

Those three stages (‘rising’, ‘packing’ and ‘gelation’)are theoretically more sensitive because the physical effectof a controlled ultrasonic irradiation effectively pumps in/outthe dissolved gas/vapour into the bubbles and, therefore,controls their final size. The ‘bouncing’ (i.e. successive,rapid stretching and shrinking of the cell walls) caused bythe ultrasonic wave contributes to an increase in the periodof viscoelastic behaviour of the forming matrix. When theacoustic pressure was above the lower threshold (figure 8),bubble growth occurred, as the driving force for expansionwas larger than the resisting force. However, in later periodsof the ‘gelation’ stage, and in the ‘solidification’ stage, theviscosity was too large for an oscillatory behaviour to occur,so the expansion halted. There seems to be a consensusin the literature that a degree of viscoelasticity enhancesbubble growth [36–38] especially in low molecular weightpolymers [39] and, as this research work assumes, in earlystages of high molecular ones. But more recent studies [40, 41]suggest that a reduction in viscoelasticity has little effect onbubble growth, and instead place more importance on blowingagent concentration, pressure surrounding the bubble and

diffusivity. This does not contradict the experimental resultsobtained here, since the nature of the effect of ultrasound onthe foaming process is not well known, and it can crediblyclaim to influence both of these phenomena at the same time:viscoelasticity (i.e. incremented shear rate due to mechanicalstirring), enhancement of diffusivity, as well as pressuregradient provoked by stable cavitation.

Due to the difficulty in testing the viscosity of a polymericmaterial near the liquid–solid transition, many authors havereported opposing views during the past 20 years. However,recent technological advances are permitting more accuratemeasurements, and the latest published results [42, 43]indicate that viscosity of the polyurethane viscoelastic meltis dependent on the shear rate and so a shear-thinningeffect (i.e. viscosity decreases when shear rate is applied,independently with time) exists. Consequently, the authorsbelieve that one of the mechanisms underpinning thephenomena reported here is that ultrasound provokes ‘stirring’and so the viscosity of the PU melt decreases due to itsviscoelastic nature and the effect of shear-thinning reducingviscosity. Although the expression used to model the variationof bubble volume in the polymeric matrix (equation (19)) isnot a complete model of the process (e.g. assumes sphericalbubble shape, a linear relationship between the matrix viscosityand the bubble surface, etc), it allows the relative importanceof the various parameters to be characterized. Further work isrequired to provide a quantitative understanding of the differentinteracting mechanisms incorporated in this expression.

6. Conclusions

During the foam polymerization reaction, the acoustic pressurein the water bath varied causing the bubbles to pulsate ina state of ‘stable cavitation’ (i.e. rectified diffusion). Thispulsation of the bubbles ‘pumped’ gas from the liquid to thegas phase inducing them to increase in volume. The eventualsolidification resulted in a porous material with a cellularstructure that reflected the acoustic field imposed upon it.

The authors conclude that, when conditions of stablecavitation are established, ultrasound can create porositygradation by producing bubbles of different sizes dependingon the acoustic pressure to which they are subjected. Thismechanism allows the engineering of standing waves to ‘tailor’the porosity of the polymeric matrix that solidifies into a foam.

The results in this work also offer a valuable insight tothe importance of the ‘packing’ and ‘gelation’ stages, andthe mechanisms that makes them ‘sensitive’ to ultrasonicirradiation. It is believed that controlled ultrasonic irradiationaffects convective mass transfer, especially during ‘rising’ and‘packing’ stages of the foaming process, and enhances thediffusion of the blowing agent (i.e. CO2 gas) from bubble tobubble in the ‘packing’ and ‘gelation’ stages.

Current work is focused on the definition of acousticproperties (i.e. acoustic impedance, Z ) which vary withreaction time and in turn will contribute to physical propertiessuch as viscosity in the foaming melt. The hope is that this willallow an even more accurate correlation of ultrasonic energylevels to values of porosity in the sonicated foams.

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The effects of ultrasound on porosity demonstrated by thiswork offer the prospect of a flexible manufacturing process thatcan control and adjust the cellular geometry of foam ‘electron-ically’ and hence ensure that the resulting characteristics of theheterogeneous material match the functional requirements inspecific applications where engineered cellular structures re-quire to be customized (e.g. biomimetics and orthopaedics;structural components, etc).

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